A000212 -id:A000212 - cp's OEIS frontend

A033440 Number of edges in 8-partite Turán graph of order n.

0, 0, 1, 3, 6, 10, 15, 21, 28, 35, 43, 52, 62, 73, 85, 98, 112, 126, 141, 157, 174, 192, 211, 231, 252, 273, 295, 318, 342, 367, 393, 420, 448, 476, 505, 535, 566, 598, 631, 665, 700, 735, 771, 808, 846, 885, 925

Offset: 0

N. J. A. Sloane

Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Turán Graph [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033441, A033442, A033443, A033444. [Reinhard Zumkeller, Nov 30 2009]

CoefficientList[Series[- x^2 (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)/((x - 1)^3 (x + 1) (x^2 + 1) (x^4 + 1)), {x, 0, 50}], x] (* Vincenzo Librandi, Oct 19 2013 *)
LinearRecurrence[{2,-1,0,0,0,0,0,1,-2,1},{0,0,1,3,6,10,15,21,28,35},50] (* Harvey P. Dale, Mar 23 2015 *)

a(n) = round( (7/16)*n(n-2) ) +0 or -1 depending on n: if there is k such 8k+4<=n<=8k+6 then a(n) = floor( (7/16)*n*(n-2)) otherwise a(n) = round( (7/16)*n(n-2)). E.g. because 8*2+4<=21<=8*2+6 a(n) = floor((7/16)*21*19) = floor(174, 5625)=174. - Benoit Cloitre, Jan 17 2002
a(n) = Sum_{k=0..n} A168181(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x^6+x^5+x^4+x^3+x^2+x+1)/((x-1)^3*(x+1)*(x^2+1)*(x^4+1)). [Colin Barker, Aug 09 2012]
a(n) = Sum_{i=1..n} floor(7*i/8). - Wesley Ivan Hurt, Sep 12 2017

A033442 Number of edges in 10-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 54, 64, 75, 87, 100, 114, 129, 145, 162, 180, 198, 217, 237, 258, 280, 303, 327, 352, 378, 405, 432, 460, 489, 519, 550, 582, 615, 649, 684, 720, 756, 793, 831, 870, 910, 951, 993, 1036, 1080, 1125, 1170, 1216, 1263

Offset: 0

N. J. A. Sloane

Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Turán Graph [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033440, A033441, A033443, A033444. [Reinhard Zumkeller, Nov 30 2009]

CoefficientList[Series[- x^2 (x^2 + x + 1) (x^6 + x^3 + 1)/((x - 1)^3 (x + 1) (x^4 - x^3 + x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 20 2013 *)

a(n) = Sum_{k=0..n} A168184(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x^2+x+1)*(x^6+x^3+1)/((x-1)^3*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)). [Colin Barker, Aug 09 2012]
a(n) = Sum_{i=1..n} floor(9*i/10). - Wesley Ivan Hurt, Sep 12 2017

More terms from Vincenzo Librandi, Oct 20 2013

A033443 Number of edges in 11-partite Turán graph of order n.

Original entry on oeis.org

0, 0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 65, 76, 88, 101, 115, 130, 146, 163, 181, 200, 220, 240, 261, 283, 306, 330, 355, 381, 408, 436, 465, 495, 525, 556, 588, 621, 655, 690, 726, 763, 801, 840, 880, 920, 961, 1003, 1046, 1090, 1135, 1181, 1228, 1276

Offset: 0

N. J. A. Sloane

Graham et al., Handbook of Combinatorics, Vol. 2, p. 1234.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Turán Graph [_Reinhard Zumkeller_, Nov 30 2009]
Wikipedia, Turán graph [_Reinhard Zumkeller_, Nov 30 2009]
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,0,0,1,-2,1).

Cf. A002620, A000212, A033436, A033437, A033438, A033439, A033440, A033441, A033442, A033444. [Reinhard Zumkeller, Nov 30 2009]

CoefficientList[Series[- x^2 (x + 1) (x^4 - x^3 + x^2 - x + 1) (x^4 + x^3 + x^2 + x + 1)/((x - 1)^3 (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)), {x, 0, 60}], x] (* Vincenzo Librandi, Oct 20 2013 *)

a(n) = Sum_{k=0..n} A145568(k)*(n-k). [Reinhard Zumkeller, Nov 30 2009]
G.f.: -x^2*(x+1)*(x^4-x^3+x^2-x+1)*(x^4+x^3+x^2+x+1)/((x-1)^3*(x^10+x^9+x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1)). [Colin Barker, Aug 09 2012]
a(n) = Sum_{i=1..n} floor(10*i/11). - Wesley Ivan Hurt, Sep 12 2017

More terms from Vincenzo Librandi, Oct 20 2013

A118013 Triangle read by rows: T(n,k) = floor(n^2/k), 1<=k<=n.

Original entry on oeis.org

1, 4, 2, 9, 4, 3, 16, 8, 5, 4, 25, 12, 8, 6, 5, 36, 18, 12, 9, 7, 6, 49, 24, 16, 12, 9, 8, 7, 64, 32, 21, 16, 12, 10, 9, 8, 81, 40, 27, 20, 16, 13, 11, 10, 9, 100, 50, 33, 25, 20, 16, 14, 12, 11, 10, 121, 60, 40, 30, 24, 20, 17, 15, 13, 12, 11, 144, 72, 48, 36, 28, 24, 20, 18, 16, 14

Offset: 1

Reinhard Zumkeller, Apr 10 2006

T(n,1) = A000290(n); T(n,n) = n;
T(n,2) = A007590(n) for n>1;
T(n,3) = A000212(n) for n>2;
T(n,4) = A002620(n) for n>3;
T(n,5) = A118015(n) for n>4;
T(n,6) = A056827(n) for n>5;
central terms give A008574: T(2*k-1,k) = 4*(k-1)+0^(k-1);
row sums give A118014.

			Triangle begins:
1,
4, 2,
9, 4, 3,
16, 8, 5, 4,

Reinhard Zumkeller, Rows n=1..100 of triangle, flattened

Cf. A010766.

a118013 n k = a118013_tabl !! (n-1) !! (k-1)
a118013_row n = map (div (n^2)) [1..n]
a118013_tabl = map a118013_row [1..]
-- Reinhard Zumkeller, Jan 22 2012

T(n,k)=n^2\k \\ Charles R Greathouse IV, Jan 15 2012

A212536 T(n,k)=Number of nondecreasing sequences of n 1..k integers with every element dividing the sequence sum.

Original entry on oeis.org

1, 2, 1, 3, 2, 1, 4, 3, 3, 1, 5, 4, 5, 3, 1, 6, 5, 7, 5, 4, 1, 7, 6, 8, 10, 8, 4, 1, 8, 7, 11, 12, 15, 8, 5, 1, 9, 8, 12, 17, 21, 15, 12, 5, 1, 10, 9, 14, 18, 30, 21, 24, 12, 6, 1, 11, 10, 16, 23, 33, 40, 33, 29, 16, 6, 1, 12, 11, 18, 26, 46, 44, 69, 40, 39, 16, 7, 1, 13, 12, 19, 30, 53, 64, 83, 91

Offset: 1

R. H. Hardin May 20 2012

Table starts
.1.2..3..4..5...6...7...8...9..10..11...12...13...14...15...16...17...18...19
.1.2..3..4..5...6...7...8...9..10..11...12...13...14...15...16...17...18...19
.1.3..5..7..8..11..12..14..16..18..19...22...23...25...27...29...30...33...34
.1.3..5.10.12..17..18..23..26..30..31...40...41...43...47...52...53...59...60
.1.4..8.15.21..30..33..46..53..66..67...87...88...95..111..125..126..143..144
.1.4..8.15.21..40..44..64..76.103.104..148..149..165..197..229..230..271..272
.1.5.12.24.33..69..83.116.145.188.193..290..293..332..428..496..497..606..607
.1.5.12.29.40..91.106.161.202.266.272..474..478..561..747..874..876.1141.1142
.1.6.16.39.57.130.157.245.331.439.455..867..878.1034.1417.1646.1651.2236.2240
.1.6.16.45.70.166.200.334.451.644.665.1424.1440.1713.2384.2785.2793.3927.3932

			Some solutions for n=8 k=4
..1....1....2....1....1....1....1....2....1....1....1....1....2....2....1....1
..1....2....2....2....1....1....1....3....1....1....1....1....2....2....2....1
..2....3....3....2....1....1....1....3....1....2....1....2....2....4....2....2
..2....3....3....2....1....1....1....3....1....2....1....2....2....4....3....2
..2....3....3....2....1....2....2....3....1....2....1....2....4....4....4....3
..4....4....3....3....2....2....2....3....1....2....1....2....4....4....4....3
..4....4....4....3....2....4....2....3....2....2....2....2....4....4....4....3
..4....4....4....3....3....4....2....4....2....2....4....4....4....4....4....3

R. H. Hardin, Table of n, a(n) for n = 1..877

Column 3 is A000212(floor((n+5)/2))

Row 3 is A106252

A087483 Row 0 of the order array of 3/2, i.e., row 0 of the transposable dispersion in A087465.

Original entry on oeis.org

1, 2, 4, 6, 9, 13, 17, 22, 28, 34, 41, 49, 57, 66, 76, 86, 97, 109, 121, 134, 148, 162, 177, 193, 209, 226, 244, 262, 281, 301, 321, 342, 364, 386, 409, 433, 457, 482, 508, 534, 561, 589, 617, 646, 676, 706, 737, 769, 801, 834, 868, 902, 937, 973, 1009, 1046, 1084

Offset: 0

Clark Kimberling, Sep 09 2003

Also, column 0 of the transposable dispersion in A087468.

Vincenzo Librandi, Table of n, a(n) for n = 0..10000
Clark Kimberling and John E. Brown, Partial Complements and Transposable Dispersions, J. Integer Seqs. 7 (2004), article 04.1.6.
Index entries for linear recurrences with constant coefficients, signature (2,-1,1,-2,1).

Cf. A053799, A087465, A087468.

A087483 := proc(n)
    1+floor((n+1)^2/3) ;
end proc:
seq(A087483(n),n=0..10) ; # R. J. Mathar, Aug 10 2017

LinearRecurrence[{2, -1, 1, -2, 1}, {1, 2, 4, 6, 9}, 100] (* Jean-François Alcover, Mar 29 2020 *)

a(n) = n + 1 - floor(n/3) + Sum_{i=1..n} floor(2i/3).
a(n) = 1 + floor((n+1)^2/3) = 1 + A000212(n+1).
a(n) = A192735(n+2) / (n+2). - Reinhard Zumkeller, Jul 08 2011
G.f.: -(x^4-x^3+x^2+1) / ((x-1)^3*(x^2+x+1)). - Colin Barker, Mar 31 2013

Edited by Max Alekseyev, Dec 05 2013

A143974 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark those having x+y=1(mod 3); then R(m,n) is the number of marked unit squares in the rectangle [0,m]x[0,n].

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 2, 1, 2, 3, 4, 4, 3, 2, 2, 4, 5, 5, 5, 4, 2, 2, 4, 6, 6, 6, 6, 4, 2, 3, 5, 7, 8, 8, 8, 7, 5, 3, 3, 6, 8, 9, 10, 10, 9, 8, 6, 3, 3, 6, 9, 10, 11, 12, 11, 10, 9, 6, 3, 4, 7, 10, 12, 13, 14, 14, 13, 12, 10, 7, 4, 4, 8, 11, 13, 15, 16, 16, 16, 15, 13, 11, 8, 4, 4, 8

Offset: 1

Clark Kimberling, Sep 06 2008

Diagonals: A000212, A128422, A032765, A143975, A071408.

Rows: A002264, A004523, A000027, A004773, A138235, A005843, A047349 et al.

			Northwest corner:
0 0 1 1 1 2
0 1 2 2 3 4
1 2 3 4 5 6
1 2 4 5 6 8
1 3 5 6 8 10
R(3,4) counts these marked squares: (1,3), (2,2), (3,1), (3,4).

Cf. A143976, A143977.

R(m,n)=floor(mn/3).

A143977 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having |x-y| == 0 (mod 3); then R(m,n) is the number of marked squares in the rectangle [0,m] X [0,n].

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 3, 3, 3, 2, 2, 4, 4, 4, 4, 2, 3, 4, 5, 6, 5, 4, 3, 3, 5, 6, 7, 7, 6, 5, 3, 3, 6, 7, 8, 9, 8, 7, 6, 3, 4, 6, 8, 10, 10, 10, 10, 8, 6, 4, 4, 7, 9, 11, 12, 12, 12, 11, 9, 7, 4, 4, 8, 10, 12, 14, 14, 14, 14, 12, 10, 8, 4, 5, 8, 11, 14, 15, 16, 17, 16, 15, 14, 11, 8, 5

Offset: 1

Clark Kimberling, Sep 06 2008

Rows numbered 3,6,9,12,15,... are, except for initial terms, multiples of (1,2,3,4,5,6,7,...) = A000027.

			Northwest corner:
  1  1  1  2  2  2  3
  1  2  2  3  4  4  5
  1  2  3  4  5  6  7
  2  3  4  6  7  8 10
  2  4  5  7  9 10 12

Stefano Spezia, First 140 antidiagonals of the array, flattened

Diagonals: A008810, A007984, A000212, A128422.
Rows and columns: A002264, A004523, A000027, A004772, A047212, et al.
Cf. A143974, A143976, A143979.

T[m_,n_]:=Ceiling[m n/3];Flatten[Table[T[m-n+1,n],{m,13},{n,m}]] (* Stefano Spezia, Oct 27 2022 *)

R(m,n) = ceiling(m*n/3). [Corrected by Stefano Spezia, Oct 27 2022]

A277646 Triangle T(n,k) = floor(n^2/k) for 1 <= k <= n^2, read by rows.

Original entry on oeis.org

1, 4, 2, 1, 1, 9, 4, 3, 2, 1, 1, 1, 1, 1, 16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 36, 18, 12, 9, 7, 6, 5, 4, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 49, 24, 16, 12, 9, 8, 7, 6

Offset: 1

Jason Kimberley, Nov 09 2016

			The first five rows of the triangle are:
1;
4, 2, 1, 1;
9, 4, 3, 2, 1, 1, 1, 1, 1;
16, 8, 5, 4, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1;
25, 12, 8, 6, 5, 4, 3, 3, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1;

Jason Kimberley, Table of n, a(n) for n = 1..10416 (the first 31 rows of the triangle)

Cf. Related triangles: A010766, A277647, A277648.
Rows of this triangle (with infinite trailing zeros):
T(1,k) = A000007(k-1),
T(2,k) = A033324(k),
T(3,k) = A033329(k),
T(4,k) = A033336(k),
T(5,k) = A033345(k),
T(6,k) = A033356(k),
T(7,k) = A033369(k),
T(8,k) = A033384(k),
T(9,k) = A033401(k),
T(10,k) = A033420(k),
T(100,k) = A033422(k),
T(10^3,k) = A033426(k),
T(10^4,k) = A033424(k).
Columns of this triangle:
T(n,1) = A000290(n),
T(n,2) = A007590(n),
T(n,3) = A000212(n),
T(n,4) = A002620(n),
T(n,5) = A118015(n),
T(n,6) = A056827(n),
T(n,7) = A056834(n),
T(n,8) = A130519(n+1),
T(n,9) = A056838(n),
T(n,10)= A056865(n),
T(n,12)= A174709(n+2).

A277646:=func;
[A277646(n,k):k in[1..n^2],n in[1..7]];

Table[Floor[n^2/k], {n, 7}, {k, n^2}] // Flatten (* Michael De Vlieger, Nov 24 2016 *)

T(n,k) = A010766(n^2,k).

A152729 a(n) = (n-2)^4 - a(n-1) - a(n-2), with a(1) = a(2) = 0.

Original entry on oeis.org

0, 0, 1, 15, 65, 176, 384, 736, 1281, 2079, 3201, 4720, 6720, 9296, 12545, 16575, 21505, 27456, 34560, 42960, 52801, 64239, 77441, 92576, 109824, 129376, 151425, 176175, 203841, 234640, 268800, 306560, 348161, 393855, 443905, 498576, 558144

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 11 2008

a(n+2) - a(n-1) = n^4 - (n-1)^4 = A005917(n) for all n in Z. - Michael Somos, Sep 02 2018

			0 + 0 + 1 = 1^4; 0 + 1 + 15 = 2^4; 1 + 15 + 65 = 3^4; ...
G.f. = x^3 + 15*x^4 + 65*x^5 + 176*x^6 + 384*x^7 + 736*x^8 + 1281*x^9 + ... - _Michael Somos_, Sep 02 2018

G. C. Greubel, Table of n, a(n) for n = 1..5000
Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

Cf. A005917, A152728, A152725, A152726, A000212.

m:=50; R:=PowerSeriesRing(Integers(), m); [0,0] cat Coefficients(R!(x^3*(x+1)*(x^2+10*x+1)/((1-x)^5*(x^2+x+1)))); // G. C. Greubel, Sep 01 2018

k0=k1=0;lst={k0,k1};Do[kt=k1;k1=n^4-k1-k0;k0=kt;AppendTo[lst,k1],{n,1,4!}];lst
LinearRecurrence[{4,-6,5,-5,6,-4,1}, {0,0,1,15,65,176,384}, 50] (* G. C. Greubel, Sep 01 2018 *)
a[ n_] := With[ {m = Max[n, 2 - n]}, SeriesCoefficient[ x^3 (1 + x) (1 + 10 x + x^2) / ((1 - x)^5 (1 + x + x^2)), {x , 0, m}]]; (* Michael Somos, Sep 02 2018 *)

concat([0,0], Vec(-x^3*(x+1)*(x^2+10*x+1)/((x-1)^5*(x^2+x+1)) + O(x^100))) \\ Colin Barker, Oct 28 2014

{a(n) = my(m = max(n, 2 - n)); polcoeff( x^3 * (1 + x) * (1 + 10*x + x^2) / ((1 - x)^5 * (1 + x + x^2)) + x * O(x^m), m)}; /* Michael Somos, Sep 02 2018 */

G.f.: -x^3*(x+1)*(x^2+10*x+1) / ((x-1)^5*(x^2+x+1)). - Colin Barker, Oct 28 2014

a(n) = a(2 - n) for all n in Z. - Michael Somos, Sep 02 2018

Definition adapted to offset by Georg Fischer, Jun 18 2021

cp's OEIS Frontend

A033440 Number of edges in 8-partite Turán graph of order n.

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A033442 Number of edges in 10-partite Turán graph of order n.

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A033443 Number of edges in 11-partite Turán graph of order n.

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A118013 Triangle read by rows: T(n,k) = floor(n^2/k), 1<=k<=n.

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A212536 T(n,k)=Number of nondecreasing sequences of n 1..k integers with every element dividing the sequence sum.

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A087483 Row 0 of the order array of 3/2, i.e., row 0 of the transposable dispersion in A087465.

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A143974 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark those having x+y=1(mod 3); then R(m,n) is the number of marked unit squares in the rectangle [0,m]x[0,n].

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A143977 Rectangular array R by antidiagonals: label each unit square in the first quadrant lattice by its northeast vertex (x,y) and mark squares having |x-y| == 0 (mod 3); then R(m,n) is the number of marked squares in the rectangle [0,m] X [0,n].

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